\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  3.0 & +  7.00 x_{1} & +  2.00 x_{2} & -10.00 x_{3} & -2.00 x_{4} & -6.00 x_{5}\\
 x_{7}   &  1.0 & -1.00 x_{1} & +  3.00 x_{2} & +  8.00 x_{3} & +  5.00 x_{4} & -4.00 x_{5}\\
 x_{8}   &  -24.0 & +  1.00 x_{1} & + 10.00 x_{2} & +  8.00 x_{3} & -8.00 x_{4} & +  9.00 x_{5}\\
 x_{9}   &  6.0 & -4.00 x_{1} & +  7.00 x_{2} & +  7.00 x_{3} & +  4.00 x_{4} & -6.00 x_{5}\\
 x_{10}   &  43.0 & -9.00 x_{1} &   & -3.00 x_{3} & -7.00 x_{4} & -4.00 x_{5}\\
\hline
z    &  0.0 & +  5.00 x_{1} & -1.00 x_{2} & +  4.00 x_{3} & +  4.00 x_{4} & +  5.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,10$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -7.00 y_{6} & +  1.00 y_{7} & -1.00 y_{8} & +  4.00 y_{9} & +  9.00 y_{10}\\
 y_{2}   &  1.0 & -2.00 y_{6} & -3.00 y_{7} & -10.00 y_{8} & -7.00 y_{9} &   \\
 y_{3}   &  1.0 & + 10.00 y_{6} & -8.00 y_{7} & -8.00 y_{8} & -7.00 y_{9} & +  3.00 y_{10}\\
 y_{4}   &  1.0 & +  2.00 y_{6} & -5.00 y_{7} & +  8.00 y_{8} & -4.00 y_{9} & +  7.00 y_{10}\\
 y_{5}   &  1.0 & +  6.00 y_{6} & +  4.00 y_{7} & -9.00 y_{8} & +  6.00 y_{9} & +  4.00 y_{10}\\
\hline
z    &  -0 & -3.00 y_{6} & -1.00 y_{7} & + 24.00 y_{8} & -6.00 y_{9} & -43.00 y_{10}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 2 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  0.9 & -6.80 y_{6} & +  1.30 y_{7} & +  0.10 y_{2} & +  4.70 y_{9} & +  9.00 y_{10}\\
 y_{8}   &  0.1 & -0.20 y_{6} & -0.30 y_{7} & -0.10 y_{2} & -0.70 y_{9} &   \\
 y_{3}   &  0.2 & + 11.60 y_{6} & -5.60 y_{7} & +  0.80 y_{2} & -1.40 y_{9} & +  3.00 y_{10}\\
 y_{4}   &  1.8 & +  0.40 y_{6} & -7.40 y_{7} & -0.80 y_{2} & -9.60 y_{9} & +  7.00 y_{10}\\
 y_{5}   &  0.1 & +  7.80 y_{6} & +  6.70 y_{7} & +  0.90 y_{2} & + 12.30 y_{9} & +  4.00 y_{10}\\
\hline
z    &  2.4 & -7.80 y_{6} & -8.20 y_{7} & -2.40 y_{2} & -22.80 y_{9} & -43.00 y_{10}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  7.8 & +  6.80 x_{1} & +  0.20 x_{8} & -11.60 x_{3} & -0.40 x_{4} & -7.80 x_{5}\\
 x_{7}   &  8.2 & -1.30 x_{1} & +  0.30 x_{8} & +  5.60 x_{3} & +  7.40 x_{4} & -6.70 x_{5}\\
 x_{2}   &  2.4 & -0.10 x_{1} & +  0.10 x_{8} & -0.80 x_{3} & +  0.80 x_{4} & -0.90 x_{5}\\
 x_{9}   &  22.8 & -4.70 x_{1} & +  0.70 x_{8} & +  1.40 x_{3} & +  9.60 x_{4} & -12.30 x_{5}\\
 x_{10}   &  43.0 & -9.00 x_{1} &   & -3.00 x_{3} & -7.00 x_{4} & -4.00 x_{5}\\
\hline
z    &  -2.4 & -0.90 x_{1} & -0.10 x_{8} & -0.20 x_{3} & -1.80 x_{4} & -0.10 x_{5}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  7.8 & +  6.80 x_{1} & +  0.20 x_{8} & -11.60 x_{3} & -0.40 x_{4} & -7.80 x_{5}\\
 x_{7}   &  8.2 & -1.30 x_{1} & +  0.30 x_{8} & +  5.60 x_{3} & +  7.40 x_{4} & -6.70 x_{5}\\
 x_{2}   &  2.4 & -0.10 x_{1} & +  0.10 x_{8} & -0.80 x_{3} & +  0.80 x_{4} & -0.90 x_{5}\\
 x_{9}   &  22.8 & -4.70 x_{1} & +  0.70 x_{8} & +  1.40 x_{3} & +  9.60 x_{4} & -12.30 x_{5}\\
 x_{10}   &  43.0 & -9.00 x_{1} &   & -3.00 x_{3} & -7.00 x_{4} & -4.00 x_{5}\\
\hline
z    &  -2.4 & +  5.10 x_{1} & -0.10 x_{8} & +  4.80 x_{3} & +  3.20 x_{4} & +  5.90 x_{5}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  7.8 & +  6.80 x_{1} & +  0.20 x_{8} & -11.60 x_{3} & -0.40 x_{4} & -7.80 x_{5}\\
 x_{7}   &  8.2 & -1.30 x_{1} & +  0.30 x_{8} & +  5.60 x_{3} & +  7.40 x_{4} & -6.70 x_{5}\\
 x_{2}   &  2.4 & -0.10 x_{1} & +  0.10 x_{8} & -0.80 x_{3} & +  0.80 x_{4} & -0.90 x_{5}\\
 x_{9}   &  22.8 & -4.70 x_{1} & +  0.70 x_{8} & +  1.40 x_{3} & +  9.60 x_{4} & -12.30 x_{5}\\
 x_{10}   &  43.0 & -9.00 x_{1} &   & -3.00 x_{3} & -7.00 x_{4} & -4.00 x_{5}\\
\hline
z    &  -2.4 & +  5.10 x_{1} & -0.10 x_{8} & +  4.80 x_{3} & +  3.20 x_{4} & +  5.90 x_{5}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{10} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  40.2888888889 & -0.76 x_{10} & +  0.20 x_{8} & -13.87 x_{3} & -5.69 x_{4} & -10.82 x_{5}\\
 x_{7}   &  1.98888888889 & +  0.14 x_{10} & +  0.30 x_{8} & +  6.03 x_{3} & +  8.41 x_{4} & -6.12 x_{5}\\
 x_{2}   &  1.92222222222 & +  0.01 x_{10} & +  0.10 x_{8} & -0.77 x_{3} & +  0.88 x_{4} & -0.86 x_{5}\\
 x_{9}   &  0.344444444444 & +  0.52 x_{10} & +  0.70 x_{8} & +  2.97 x_{3} & + 13.26 x_{4} & -10.21 x_{5}\\
 x_{1}   &  4.77777777778 & -0.11 x_{10} &   & -0.33 x_{3} & -0.78 x_{4} & -0.44 x_{5}\\
\hline
z    &  21.9666666667 & -0.57 x_{10} & -0.10 x_{8} & +  3.10 x_{3} & -0.77 x_{4} & +  3.63 x_{5}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  5.52173913043 & -0.96 x_{10} & -1.61 x_{8} & + 18.09 x_{2} & -21.57 x_{4} & +  4.65 x_{5}\\
 x_{7}   &  17.115942029 & +  0.23 x_{10} & +  1.09 x_{8} & -7.87 x_{2} & + 15.32 x_{4} & -12.86 x_{5}\\
 x_{3}   &  2.50724637681 & +  0.01 x_{10} & +  0.13 x_{8} & -1.30 x_{2} & +  1.14 x_{4} & -1.12 x_{5}\\
 x_{9}   &  7.78260869565 & +  0.57 x_{10} & +  1.09 x_{8} & -3.87 x_{2} & + 16.65 x_{4} & -13.52 x_{5}\\
 x_{1}   &  3.94202898551 & -0.12 x_{10} & -0.04 x_{8} & +  0.43 x_{2} & -1.16 x_{4} & -0.07 x_{5}\\
\hline
z    &  29.7391304348 & -0.52 x_{10} & +  0.30 x_{8} & -4.04 x_{2} & +  2.78 x_{4} & +  0.17 x_{5}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.256048387097 & -0.04 x_{10} & -0.07 x_{8} & +  0.84 x_{2} & -0.05 x_{6} & +  0.22 x_{5}\\
 x_{7}   &  21.0383064516 & -0.45 x_{10} & -0.06 x_{8} & +  4.98 x_{2} & -0.71 x_{6} & -9.55 x_{5}\\
 x_{3}   &  2.80040322581 & -0.04 x_{10} & +  0.05 x_{8} & -0.34 x_{2} & -0.05 x_{6} & -0.87 x_{5}\\
 x_{9}   &  12.0463709677 & -0.17 x_{10} & -0.16 x_{8} & + 10.10 x_{2} & -0.77 x_{6} & -9.93 x_{5}\\
 x_{1}   &  3.64516129032 & -0.06 x_{10} & +  0.04 x_{8} & -0.54 x_{2} & +  0.05 x_{6} & -0.32 x_{5}\\
\hline
z    &  30.4516129032 & -0.65 x_{10} & +  0.10 x_{8} & -1.71 x_{2} & -0.13 x_{6} & +  0.77 x_{5}\\
\end{array}\]


 $ x_{5} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.517766497462 & -0.05 x_{10} & -0.08 x_{8} & +  1.06 x_{2} & -0.06 x_{6} & -0.02 x_{9}\\
 x_{7}   &  9.45177664975 & -0.28 x_{10} & +  0.09 x_{8} & -4.73 x_{2} & +  0.03 x_{6} & +  0.96 x_{9}\\
 x_{3}   &  1.7461928934 & -0.02 x_{10} & +  0.06 x_{8} & -1.23 x_{2} & +  0.01 x_{6} & +  0.09 x_{9}\\
 x_{5}   &  1.21319796954 & -0.02 x_{10} & -0.02 x_{8} & +  1.02 x_{2} & -0.08 x_{6} & -0.10 x_{9}\\
 x_{1}   &  3.2538071066 & -0.06 x_{10} & +  0.05 x_{8} & -0.87 x_{2} & +  0.08 x_{6} & +  0.03 x_{9}\\
\hline
z    &  31.3908629442 & -0.66 x_{10} & +  0.08 x_{8} & -0.92 x_{2} & -0.19 x_{6} & -0.08 x_{9}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  6.640625 & -0.62 x_{10} & -12.83 x_{4} & + 13.57 x_{2} & -0.81 x_{6} & -0.28 x_{9}\\
 x_{7}   &  10.0729166667 & -0.34 x_{10} & -1.20 x_{4} & -3.46 x_{2} & -0.04 x_{6} & +  0.94 x_{9}\\
 x_{3}   &  2.13541666667 & -0.06 x_{10} & -0.75 x_{4} & -0.43 x_{2} & -0.03 x_{6} & +  0.07 x_{9}\\
 x_{5}   &  1.109375 & -0.01 x_{10} & +  0.20 x_{4} & +  0.80 x_{2} & -0.07 x_{6} & -0.10 x_{9}\\
 x_{1}   &  3.57291666667 & -0.09 x_{10} & -0.62 x_{4} & -0.21 x_{2} & +  0.04 x_{6} & +  0.02 x_{9}\\
\hline
z    &  31.953125 & -0.71 x_{10} & -1.09 x_{4} & +  0.23 x_{2} & -0.26 x_{6} & -0.10 x_{9}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  46.1067669173 & -1.94 x_{10} & -17.53 x_{4} & -3.92 x_{7} & -0.98 x_{6} & +  3.39 x_{9}\\
 x_{2}   &  2.90827067669 & -0.10 x_{10} & -0.35 x_{4} & -0.29 x_{7} & -0.01 x_{6} & +  0.27 x_{9}\\
 x_{3}   &  0.878195488722 & -0.02 x_{10} & -0.60 x_{4} & +  0.12 x_{7} & -0.03 x_{6} & -0.05 x_{9}\\
 x_{5}   &  3.44962406015 & -0.09 x_{10} & -0.08 x_{4} & -0.23 x_{7} & -0.08 x_{6} & +  0.12 x_{9}\\
 x_{1}   &  2.95187969925 & -0.07 x_{10} & -0.54 x_{4} & +  0.06 x_{7} & +  0.04 x_{6} & -0.04 x_{9}\\
\hline
z    &  32.6120300752 & -0.73 x_{10} & -1.16 x_{4} & -0.07 x_{7} & -0.26 x_{6} & -0.04 x_{9}\\
\end{array}\]
Final Dictionary
Solution:  32.6120300752
Num Pivots:  7
\end{document}
